A block of mass $M$ placed inside a box descends vertically with acceleration $a$ . The block exerts a force equal
to one-fourth of its weight on the floor of the box. The value of $'a'$ will be
a.
$g$
b.
$\frac{3g}{4}$
Correct Answer
c.
$\frac{g}{2}$
d.
$\frac{g}{4}$
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Q2mcq
A block of mass $40\, kg$ slides over a surface, when a mass of $4\, kg$ is suspended through an inextensible
massless string passing over frictionless pulley as shown below.
The coefficient of kinetic friction between the surface and block is $0. 02$ . The acceleration of block is: (Given
$g= 10\,m\,s^{-2}$ .)
a.
$\frac{8}{11}\,m\,s^{-2}$
Correct Answer
b.
$1\,m\,s^{-2}$
c.
$\frac{1}{5} \,m\,s^{-2}$
d.
$\frac{4}{5} \,m\,s^{-2}$
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Q3mcq
If the electric potential at any point $(x, \,y, \,z)\,m$ in space is given by $V = 3x^{2}$ volt. The electric field at the
point $(1, \,0, \,3)\,m$ will be :
a.
$3\, Vm^{-1}$ , directed along positive $x$ -axis.
b.
$3\, Vm^{-1}$ , directed along negative $x$ -axis.
c.
$6\, Vm^{-1}$ , directed along negative $x$ -axis.
Correct Answer
d.
$6\, Vm^{-1}$ , directed along positive $x$ -axis.
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Q4mcq
Two long current carrying conductors are placed parallel to each other at a distance of $8\, cm$ between them.
The magnitude of magnetic field produced at mid-point between the two conductors due to current flowing in
them is $300\,$ $\mu$ $T$ . The equal current flowing in the two conductors is :
a.
$30\,A$ in the same direction.
b.
$60\,A$ in the opposite direction.
c.
$30\,A$ in the opposite direction. $300\,A$ in the opposite direction.
Correct Answer
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Q5nat
In a double slit experiment with monochromatic light, fringes are obtained on a screen placed at some distance
from the plane of slits. If the screen is moved by $5 \times 10^{-2}\,m$ towards the slits, the change in fringe width is
$3 \times 10^{-3}\, cm$ . If the distance between the slits is $1\, mm$ , then the wavelength of the light will be____ $nm$ .
✓ Accepted Answer:600
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Q6nat
A potential barrier of $0. 4\,V$ exists across a p-n junction. An electron enters the junction from the $n$ -side with a
speed of $6. 0 \times 10^{5}\,m\,s^{-1}$ . The speed with which electron enters the $p$ side will be $\frac{x}{3} \times 10^{5}\,m\,s^{-1}$ , then the
value of $x$ is _____.: (Given mass of electron $= 9 \times 10^{-31}\, kg$ , charge on electron $= 1. 6 \times 10^{-19}\,C$ .)
✓ Accepted Answer:14
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Q7mcq
A person can throw a ball upto a maximum range of $100\,m$ . How high above the ground he can throw the same
ball?
a.
$25\,m$
b.
$50\,m$
Correct Answer
c.
$100\,m$
d.
$200\,m$
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Q8mcq
In the following nuclear reaction,
$D \rightarrow \rightarrow^{\alpha } D_{1} \rightarrow \rightarrow^{\beta ^{-}} D_{2}^{\alpha } D_{3} \rightarrow \rightarrow^{\gamma } D_{4}$
Mass number of $\vec{D}$ is $182$ and atomic number is $74$ . Mass number and atomic number of $D_{4}$ respectively will
be
a.
$174$ and $71$
Correct Answer
b.
$174$ and $69$
c.
$172$ and $69$
d.
$172$ and $71$
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Q9mcq
A capacitor is discharging through a resistor $R$ . Consider in time $t_{1}$ , the energy stored in the capacitor reduces
to half of its initial value and in time $t_{2}$ , the charge stored reduces to one eighth of its initial value. The ratio $\frac{t_{1}}{t_{2}}$
will be
a.
$\frac{1}{2}$
b.
$\frac{1}{3}$
c.
$\frac{1}{4}$
d.
$\frac{1}{6}$
Correct Answer
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Q10nat
An inductor of $0. 5\, mH$ , a capacitor of $200\,$ $\mu$ $F$ and a resistor of $2\,\Omega$ are connected in series with a $220\,V$ ac
source. If the current is in phase with the emf, the frequency of ac source will be ______ $\times 10^{2}\, Hz$ .
✓ Accepted Answer:5
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Given below are two statements :
Statement $I$ : The electric force changes the speed of the charged particle and hence changes its kinetic energy;
whereas the magnetic force does not change the kinetic energy of the charged particle.
Statement $II$ : The electric force accelerates the positively charged particle perpendicular to the direction of
electric field. The magnetic force accelerates the moving charged particle along the direction of magnetic field.
In the light of the above statements, choose the most appropriate answer from the options given below :
a.
Both Statement $I$ and Statement $II$ are correct.
b.
Both Statement $I$ and Statement $II$ are incorrect.
c.
Statement $I$ is correct but Statement $II$ is
incorrect.
d.
Statement $I$ is incorrect but Statement $II$ is
correct.
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The displacement current of $4. 425\,$ $\mu$ $A$ is developed in the space between the plates of parallel plate capacitor
when voltage is changing at a rate of $10^{6}\,V\,s^{-1}$ . The area of each plate of the capacitor is $40\, cm^{2}$ . The
distance between each plate of the capacitor is $x\times 10^{-3}\,m$ . The value of $x$ is ,
(Permittivity of free space, $\epsilon _{0} = 8. 85 \times 10^{-12}\,C^{2}\,N^{-1}\,m^{-2}$ ) _______
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A small toy starts moving from the position of rest under a constant acceleration. If it travels a distance of $10\,m$
in $t\,s$ , the distance travelled by the toy in the next $t\,s$ will be:
a.
$10\,m$
b.
$20\,m$
c.
$30\,m$
d.
$40\,m$
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The motion of a simple pendulum executing S.H.M. is represented by the following equation
$y=Asin(\pi t+\phi )$ , where time is measured in second. The length of pendulum is
a.
$97. 23\, cm$
b.
$25. 3\, cm$
c.
$99. 4\, cm$
d.
$406. 1\, cm$
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The electric field at a point associated with a light wave is given by
$E = 200[sin(6 \times 10^{15})t+ sin(9 \times 10^{15})t]\, Vm^{-1}$
Given: $h= 4. 14 \times 10^{-15}\, eVs$
If this light falls on a metal surface having a work function of $2. 50\, eV$ , the maximum kinetic energy of the
photoelectrons will be
a.
$1. 90\, eV$
b.
$3. 27\, eV$
c.
$3. 60\, eV$
d.
$3. 42\, eV$
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The Vernier constant of Vernier callipers is $0. 1\, mm$ and it has zero error of $(-0. 05\, cm)$ . While measuring
diameter of a sphere, the main scale reading is $1. 7\, cm$ and coinciding vernier division is $5$ . The corrected
diameter will be _____ $\times 10^{-2}\,\, cm$ .
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Two point charges $Q$ each are placed at a distance $d$ apart. A third point charge $q$ is placed at a distance $x$ from
mid-point on the perpendicular bisector. The value of $x$ at which charge $q$ will experience the maximum
Coulomb's force is:
a.
$d$
b.
$\frac{d}{2}$
c.
$\frac{d}{\sqrt{2}}$
d.
$\frac{d}{2\sqrt{2}}$
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The moment of inertia of a uniform thin rod about a perpendicular axis passing through one end is $I_{1}$ . The
same rod is bent into a ring and its moment of inertia about a diameter is $I_{2}$ . If $\frac{I}{I_{2}}^{1}$ is $\frac{x\pi ^{2}}{3}$ , then the value of $x$
will be ______.
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The speed of light in media $'A'$ and $'B'$ are $2. 0 \times 10^{10}\, cm \,s^{-1}$ and $1. 5 \times 10^{10}\, cm \,s^{-1}$ respectively. A ray of
light enters from the medium $B$ to $A$ at an incident angle $'\theta '$ . If the ray suffers total internal reflection, then
a.
$\theta = sin^{-1}( \frac{4}{3})$
b.
$\theta > sin^{-1}(\frac{2}{3} )$
c.
$\theta > sin^{-1}( \frac{3}{4})$
d.
$\theta < sin^{-1}(\frac{3}{4} )$
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The combination of two identical cells, whether connected in series or parallel combination provides the same
current through an external resistance of $2\,\Omega$ . The value of internal resistance of each cell is
a.
$2\,\Omega$
b.
$4\,\Omega$
c.
$6\,\Omega$
d.
$8\,\Omega$
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The time period of a satellite revolving around earth in a given orbit is $7$ hours. If the radius of orbit is increased
to three times its previous value, then approximate new time period of the satellite will be
a.
$36\, hours$
b.
$40\, hours$
c.
$30\, hours$
d.
$25\, hours$
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The half life of a radioactive substance is $5$ years. After $x$ years a given sample of the radioactive substance
gets reduced to $6. 25%$ of its initial value. The value of $x$ is ____
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A small spherical ball of radius $0. 1\, mm$ and density $10^{4}\, kg \,m^{-3}$ falls freely under gravity through a distance
$h$ before entering a tank of water. If, after entering the water the velocity of ball does not change and it
continue to fall with same constant velocity inside water, then the value of $h$ will be ____ $m$ . (Given
$g= 10\,m\,s^{-2}$ , viscosity of water $= 1. 0 \times 10^{-5}\,N$ ‐ $s\,m^{-2}$ ).
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The TV transmission tower at a particular station has a height of $125\,m$ . For doubling the coverage of its
range, the height of the tower should be increased by
a.
$500\,m$
b.
$375\,m$
c.
$250\,m$
d.
$125\,m$
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Starting with the same initial conditions, an ideal gas expands from volume $V_{1}$ to $V_{2}$ in three different ways.
The work done by the gas is $W_{1}$ if the process is purely isothermal, $W_{2}$ , if the process is purely adiabatic and
$W_{3}$ if the process is purely isobaric. Then, choose the correct option
a.
$W_{2} <W_{1} <W_{3}$
b.
$W_{1} <W_{2} <W_{3}$
c.
$W_{2} <W_{3} <W_{1}$
d.
$W_{3} <W_{1} <W_{2}$
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Two resistors are connected in series across a battery as shown in figure. If a voltmeter of resistance $2000\,\Omega$ is
used to measure the potential difference across $500\,\Omega$ resister, the reading of the voltmeter will be _____ $V$
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In an experiment to determine the velocity of sound in air at room temperature using a resonance tube, the first
resonance is observed when the air column has a length of $20. 0\, cm$ for a tuning fork of frequency $400\, Hz$ is
used. The velocity of the sound at room temperature is $336\,m\,s^{-1}$ . The third resonance is observed when the
air column has a length of _____ $cm$
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At what temperature a gold ring of diameter $6. 230\, cm$ be heated so that it can be fitted on a wooden bangle of
diameter $6. 241\, cm$ ? Both the diameters have been measured at room temperature $(27^{\circ}C)$ .
(Given: coefficient of linear thermal expansion of gold $\alpha _{L} = 1. 4 \times 10^{-5}\,K^{-1}$ )
a.
$125. 7^{\circ}C$
b.
$91. 7^{\circ}C$
c.
$425. 7^{\circ}C$
d.
$152. 7^{\circ}C$
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$4. 0$ moles of argon and $5. 0$ moles of $PCl_{5}$ are introduced into an evacuated flask of $100$ litre capacity at
$610\,K$ . The system is allowed to equilibrate. At equilibrium, the total pressure of mixture was found to be
$6. 0\, atm$ . The $K_{p}$ for the reaction is [Given : $R = 0. 082\,L$ atm $K^{-1}\, mol^{-1}$ ]
a.
$2. 25$
b.
$6. 24$
c.
$12. 13$
d.
$15. 24$
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A box contains $0. 90\,g$ of liquid water in equilibrium with water vapour at $27^{\circ}C$ . The equilibrium vapour
pressure of water at $27^{\circ}C\,32. 0$ Torr. When the volume of the box is increased, some of the liquid water
evaporates to maintain the equilibrium pressure. If all the liquid water evaporates, then the volume of the box
must be ______ litre. [nearest integer]
(Given: $R = 0. 082\,L\, atm \,K^{-1}\, mol^{-1}$ _
(Ignore the volume of the liquid water and assume water vapours behave as an ideal gas.)
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$42. 12%$ (w/v) solution of $NaCl$ causes precipitation of a certain sol in $10$ hours. The coagulating value of
$NaCl$ for the sol is
[Given : Molar mass : $Na = 23. 0\,g\, mol^{-1}; \,Cl = 35. 5\,g\, mol^{-1}$ ]
a.
$36\, mmolL^{-1}$
b.
$36\, mol \,L^{-1}$
c.
$1440\, mol \,L^{-1}$
d.
$1440\,m\, molL^{-1}$
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Given below are two statements .
Statement $I$ : In $CuSO_{4}\cdot 5H_{2}O, \,Cu -O$ bonds are present.
Statement $II$ : In $CuSO_{4}. 5H_{2}O$ , ligands coordinating with $Cu \,(II)$ ion are $O$ -and $S$ -based ligands.
In the light of the above statements, choose the correct answer from the options given below
a.
Both Statement $I$ and Statement $II$ are correct
b.
Both Statement $I$ and Statement $II$ are incorrect
c.
Statement $I$ is correct but Statement $II$ is
incorrect
d.
Statement $I$ is incorrect but Statement $II$ is
correct
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Match List $I$ with List $II$ .
List $I$ With Ore List $II$ Composition
A.Siderite I. $FeCO_{3}$
B.Malachite II. $CuCO_{3}\cdot Cu(OH)_{2}$
C.Sphalerite III. $ZnS$
D.Calamine IV. $ZnCO_{3}$
Choose the correct answer from the options given below:
a.
A-I, B-II, C-III, D-IV
b.
A-III, B-IV, C-E, D-I
c.
A-IV, B-III, C-I, D-II
d.
A-I, B-II, C-IV, D-III
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For the reaction given below:
$CoCl_{3}. \, xNH_{3}+ AgNO_{3}(aq) \rightarrow \rightarrow$
If two equivalents of $AgCl$ precipitate out, then the value of $x$ will be _______
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Sulphur dioxide is one of the components of polluted air. $SO_{2}$ is also a major contributor to acid rain. The
correct and complete reaction to represent acid rain caused by $SO_{2}$ is :
Elevation in boiling point for $1. 5$ molal solution of glucose in water is $4\,K$ . The depression in freezing point
for $4. 5$ molal solution of glucose in water is $4\,K$ . The ratio of molal elevation constant to molal depression
constant $(K_{b}/K_{f})$ is
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Two isomers $(A)$ and $(B)$ with Molar mass $184\,g/ mol$ and elemental compositio $C, \,52. 2%; \,H, \,49%$ and
$Br \,42. 9%$ gave benzoic acid and $p-$ bromobenzoic acid, respectively on oxidation with $KMnO_{4}$ . Isomer $'A'$
is optically active and gives a pale yellow precipitate when warmed with alcoholic $AgNO_{3}$ . Isomer $'A'$ and $'B'$
are, respectively :
c.
$H_{3}C -CHBr -C_{6}H_{5}$ and
d.
and $H_{3}C -CHBr -C_{6}H_{5}$
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The equation $k =(6. 5 \times 10^{12}\,s^{-1})e^{-26000\,K/T}$ is followed for the decomposition of compound $A$ . The
activation energy for the reaction is ____ $KJ \, mol^{-1}$ . [nearest integer]
(Given: $R = 8. 314\,J\,K^{-1}\, mol^{-1}$ )
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A white precipitate was formed when $BaCl_{2}$ was added to water extract of an inorganic salt. Further, a gas $'X'$
with characteristic odour was released when the formed white precipitate was dissolved in dilute HCI. The
anion present in the inorganic salt is:
a.
$I^{-}$
b.
$SO_{3}^{2-}$
c.
$S^{2-}$
d.
$NO^{-}_{2}$
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$2. 2\,g$ of nitrous oxide $(N_{2}O)$ gas is cooled at a constant pressure of $1$ atm from $310\,K$ to $270\,K$ causing the
compression of the gas from $217. 1\, mL$ to $167. 75\, mL$ . The change in internal energy of the process, $\triangle U$ is
$'-x'J$ . The value of $'x'$ is _____.
[nearest integer]
(Given: atomic mass of $N = 14\,g\, mol^{-1}$ and of $O = 16\,g\, mol^{-1}$ . Molar heat capacity of $N_{2}O$ is
$100\, JK^{-1} \, mol^{-1}$ )
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Which of the following is the correct plot for the probability density ψ $^{2}(r)$ as a function of distance $'r'$ of the
electron form the nucleus for $2s$ orbital?
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Given below are two statements: one is labelled as
Assertion $A$ and the other is labelled as Reason $R$ .
Assertion $A$ : Dacron is an example of polyester polymer.
Reason $R$ : Dacron is made up of ethylene glycol and terephthalic acid monomers.
In the light of the above statements, choose the most appropriate answer from the options given below.
a.
Both $A$ and $R$ are correct and $R$ is the
correct explanation of $A$
b.
Both $A$ and $R$ are correct but $R$ is NOT
the correct explanation of $A$ .
c.
$A$ is correct but $R$ is not correct. $A$ is not correct but $R$ is correct.
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$|\parallel |$
The cell potential for the given cell at $298\,K\, Pt|H_{2}(g, 1\, bar)| H^{+}(aq) Cu^{2+}(aq) Cu(s)$ is $0. 31\,V$ . The $pH$
of the acidic solution is found to be $3$ , whereas the concentration of $Cu^{2+}$ is $10^{x}\,M$ . The value of $x$ is
_________.
(Given: $E^{Θ}_{Cu^{2+}/ Cu} = 0. 34\,V$ and $\frac{2.303 RT}{F} = 0. 06\,V$ )
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Given below are two statements. One is labelled as
Assertion $A$ and the other is labelled as Reason $R$ .
Assertion $A$ : The first ionization enthalpy for oxygen is lower than that of nitrogen.
Reason $R$ : The four electrons in $2p$ orbitals of oxygen experience more electron-electron repulsion.
In the light of the above statements, choose the correct answer from the options given below.
a.
Both $A$ and $R$ are correct and $R$ is the correct Both $A$ and $R$ are correct but $R$ is NOT the
explanation of $A$ . correct explanation of $A$ .
c.
$A$ is correct but $R$ is not correct.
d.
$A$ is not correct but $R$ is correct
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Let $y=y(x), \,x> 1$ , be the solution of the differential equation $(x-1)\frac{dy}{dx} + 2xy= \frac{1}{x-1}$ , with $y(2) = \frac{1+e^{4}}{2e^{4}}$ .
If $y(3) = \frac{e^{\alpha }+1}{\beta e^{\alpha }}$ . then the value of $\alpha +\beta$ is equal to ______.
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Let $f$ be a real valued continuous function on $[0, \,1]$ and $f(x) =x+\int _{0}^{1}(x- t)f(t)dt$ . Then which of
the following points $(x, \,y)$ lies on the curve $y=f(x)$ ?
a.
$(2, \,4)$
b.
$(1, \,2)$
c.
$(4, \,17)$
d.
$(6, \,8)$
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Let $P :y^{2} = 4ax, \,a> 0$ be a parabola with focus $S$ .Let the tangents to the parabola $P$ make an angle of $\frac{\pi}{4}$
with the line $y= 3x+ 5$ touch the parabola $P$ at $A$ and $B$ . Then the value of $a$ for which $A, \,B$ and $S$ are
collinear is:
a.
$8$ only $2$ only
c.
$\frac{1}{4}$ only
d.
any $a> 0$
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Let $f : \,R\rightarrow \rightarrow R$ be a function defined by $f(x) = (x-3)^{n_{1}}(x-5)^{n_{2}}, \,n_{1}, \,n_{2} \in N$ . The, which of
the following is NOT true?
a.
For $n_{1} = 3, \,n_{2} = 4$ , there exists $\alpha \in (3, \,5)$
where $f$ attains local maxima.
b.
For $n_{1} = 4, \,n_{2} = 3$ , there exists $\alpha \in (3, \,5)$
where $f$ attains local maxima.
c.
For $n_{1} = 3, \,n_{2} = 5$ , there exists $\alpha \in (3, \,5)$
where $f$ attains local maxima.
d.
For $n_{1} = 4, \,n_{2} = 6$ , there exists $\alpha \in (3, \,5)$
where $f$ attains local maxima.
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$2 -1_{5}_{5}$
Let $A=( )$ . If $B=I-^{5}C (adj \,A) +^{5}C (adj \,A)^{2} -\,. . .-\, C_{5}(adj \,A)$ , then the sum of
$0 2^{1}^{2}$
all elements of the matrix $B$ is:
a.
$-5$
b.
$-6$
c.
$-7$
d.
$-8$
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From the base of a pole of height $20$ meter, the angle of elevation of the top of a tower is $60^{\circ}$ . The pole
subtends an angle $30^{\circ}$ at the top of the tower. Then the height of the tower is:
a.
$15\sqrt{3}$
b.
$20\sqrt{3}$
c.
$20 + 10\sqrt{3}$
d.
$30$
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If $y=y(x)$ is the solution of the differential equation $(1 +e^{2x}) \frac{dy}{dx} + 2(1 +y^{2})e^{x} = 0$ and $y(0) = 0$ , then
$6(y'(0) +(y(log_{c}\sqrt{3}))^{2})$ is equal to:
a.
$2$
b.
$-2$
c.
$-4$
d.
$-1$
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The distance of the origin from the centroid of the triangle whose two sides have the equations $x-2y+ 1 = 0$
and $2x- y-1 = 0$ and whose orthocenter is $( \frac{7}{3}, \, \frac{7}{3})$ is:
a.
$\sqrt{2}$
b.
$2$
c.
$2\sqrt{2}$
d.
$4$
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$0 -\alpha _{49}$
Let $M =[ ]$ , where $\alpha$ is a non-zero real number and $N =\sum M^{2k}$ . If $(I- M^{2})N = -2I$ , then the
$\alpha 0^{k=1}$
positive integral value of $\alpha$ is ______.
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Let $A, \,B, \,C$ be three points whose position vectors respectively are:
$\vec{a}= ˆi+ 4ˆj+ 3ˆk$
$\vec{b}= 2ˆi+\alpha ˆj+ 4ˆk, \,\alpha \in R$
$\vec{c}= 3ˆi-2ˆj+ 5ˆk$
If $\alpha$ is the smallest positive integer for which $\vec{a}, \,\vec{b}, \,\vec{c}$ are non-collinear, then the length of the median, $\triangle ABC$ ,
through $A$ is:
a.
$\frac{\sqrt{82}}{2}$
b.
$\frac{\sqrt{62}}{2}$
c.
$\frac{\sqrt{69}}{2}$
d.
$\frac{\sqrt{66}}{2}$
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Let $f(x)$ and $g(x)$ be two real polynomials of degree $2$ and $1$ respectively. If $f(g(x)) = 8x^{2} -2x$ , and
$g(f(x)) = 4x^{2} + 6x+ 1$ , then the value of $f(2) +g(2)$ is ______.
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$15$
Let the coefficients of $x^{-1}$ and $x^{-3}$ in the expansion of $(2x\frac{1}{5} - \frac{1}{51}) , \,x> 0$ , be $m$ and $n$ respectively. If $r$
$x$
is a positive integer such $mn^{2} =^{15}C_{r.}2^{r}$ , then the value of $r$ is equal to ______.
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Let $3, \,6, \,9, \,12, \, …$ upto $78$ terms and $5, \,9, \,13, \,17, \, …$ upto $59$ terms be two series. Then, the sum of the
terms common to both the series is equal to ______.
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Let a triangle $ABC$ be inscribed in the circle $x^{2} - \sqrt{2}(x+y) +y^{2} = 0$ such that $\angle BAC = \frac{\pi}{2}$ . If the length of
side $AB$ is $\sqrt{2}$ , then the area of the $\triangle ABC$ is equal to:
a.
$1$
b.
$\frac{(\sqrt{6}+\sqrt{3})}{2}$
c.
$\frac{(\sqrt{3}+\sqrt{3})}{2}$
d.
$\frac{(\sqrt{6}+2\sqrt{3})}{4}$
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The sum of the infinite series $1 + \frac{5}{6} + \frac{12}{6^{2}} + \frac{22}{6^{3}} + \frac{35}{6^{4}} + \frac{51}{6^{5}} + \frac{70}{6^{6}} + …$ is equal to:
a.
$\frac{425}{216}$
b.
$\frac{429}{216}$
c.
$\frac{288}{125}$
d.
$\frac{280}{125}$
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Let $f$ and $g$ be twice differentiable even functions on $(-2, \,2)$ such that $f(\frac{1}{4} )= 0, \,f(\frac{1}{2})= 0, \,f(1) = 1$ and
$g(\frac{3}{4})= 0,g(1) = 2$ Then, the minimum number of solutions of $f(x)g''(x) +f'(x)g''(x) = 0$ in $(-2, \,2)$ is
equal to _____.
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Let $\frac{x-2}{3} = \frac{y+1}{-2} = \frac{z+3}{-1}$ lie on the plane $px- qy+z= 5$ , for some $p, \,q\in R$ . The shortest distance of the
plane from the origin is:
a.
$\sqrt{\frac{3}{109}}$
b.
$\sqrt{\frac{5}{142}}$
c.
$\sqrt{\frac{5}{71}}$
d.
$\sqrt{\frac{1}{142}}$
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For real numbers $a, \,b(a>b> 0)$ , let
Area ${(x, \,y) :x^{2} +y^{2} \leq a^{2}$ and $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} \geq 1}= 30\pi$
and
Area ${(x, \,y) :x^{2} +y^{2} \geq b^{2}$ and $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} \leq 1}= 18\pi$
Then the value of $(a- b)^{2}$ is equal to _____.
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$\rightarrow \rightarrow$
Let $\vec{a}= ˆi-2ˆj+ 3ˆk, \vec{b}= ˆi+ ˆj+ ˆk$ and $\vec{c}$ be a vector such that $\vec{a}\times (\vec{b}+ \vec{c})= 0$ , then the value of $3(\vec{c}. \vec{a})$ is
equal to _______.
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Let $Q$ be the mirror image of the point $P(1, \,2, \,1)$ with respect to the plane $x+ 2y+ 2z= 16$ . Let $T$ be a
plane passing through the point $Q$ and contains the line $\vec{r}= -ˆk+\lambda (ˆi+ ˆj+ 2ˆk), \,\lambda \in R$ . Then, which of the
following points lies on $T$ ?
a.
$(2, \,1, \,0)$
b.
$(1, \,2, \,1)$
c.
$(1, \,2, \,2)$
d.
$(1, \,3, \,2)$
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